##
**Rational approximations to algebraic numbers.**
*(English)*
Zbl 0064.28501

Mathematika 2, 1-20 (1955); corrigendum 2, 168 (1955).

This important paper contains a proof of the long conjectured theorem:

“If \(\alpha\) is an algebraic irrational number, and if there are infinitely many fractions \(h/q\) with \(\vert \alpha - h/q\vert \le q^{-\kappa}\), \(q > 0\), \((h, q) = 1\), then \(\kappa\le 2\).”

Much of the proof runs on classical lines. The new ideas are concerned with the multiple zeros at rational points of polynomials in many variables. If \(R(x_1,\ldots,x_m)\not\equiv 0\) is such a polynomial, and if \(\alpha_1,\ldots, \alpha_m\) are real numbers and \(r_1,\ldots, r_m\) are positive integers, write

\[ R(\alpha_1+ y_1, \ldots, \alpha_m+ y_m) = \sum_{j_1=0}^\infty \cdots \sum_{j_m=0}^\infty c(j_1,\ldots, j_m) y_1^{j_1} \cdots y_m^{j_m}. \]

Then the index of \(R\) at \((\alpha_1,\ldots, \alpha_m)\) relative to \(r_1,\ldots, r_m\) is defined as \[ \min (j_1/r_1 + \ldots + j_m/r_m) \] extended over all \(j_1,\ldots, j_m\) with \(c(j_1,\ldots, j_m) \ne 0\). This index is a non-Archimedean (logarithmic) valuation. Next let \(B\), \(r_1,\ldots, r_m\), \(q_1,\ldots, q_m\) be positive integers, and let \(\mathfrak R_m = \mathfrak R_m(B; r_1,\ldots,r_m)\) be the set of all polynomials

\[ R(x_1,\ldots,x_m) = \sum_{k_1=0}^{r_1}\cdots \sum_{k_m=0}^{r_m} a_{k_1\cdots k_m} x_1^{k_1} \cdots x_1^{k_1} \not\equiv 0 \]

with integral coefficients satisfying \(\vert a_{k_1\cdots k_m}\vert \le B\). Denote by \(\Theta_m(B; q_1,\ldots,q_m; r_1,\ldots,r_m)\) the upper bound of the index of \(R\) at the point \((h_1/q_1, \ldots, h_m/q_m)\), extended over all \(R\in\mathfrak R_m\) and over all integers \(h_1,\ldots, h_m\) prime to \(q_1,\ldots, q_m\), respectively. The main lemma states:

“Let \(m\), \(r_1,\ldots, r_m\) , \(q_1,\ldots, q_m\) be positive integers and \(\delta\) a real number such that \(0 < \delta <1/m\), \(r_m > 10/\delta\), \(r_{j-1}/r_j> 1/\delta\) for \(j = 2, 3,\ldots, m\), \(\log q_1 > m (2m + 1)/\delta\), \(r_j\log q_j \ge r_1 \log q_1\) for \(j = 2, 3,\ldots, m\). Then

\[ \Theta_m\left(q_1^{\delta r_1}, q_1,\ldots, q_m; r_1,\ldots, r_m\right) < 10^m \delta^{(1/2)^m}.'' \]

This lemma is proved by induction for \(m\).

If the theorem were false, one could select \(m\) solutions \(h_j/q_j\) of \(\vert \alpha - h_j/q_j\vert < q_j^{-\kappa}\), where \(\kappa > 2\), and construct a polynomial \(R\ne 0\) with not too large integral coefficients, which would be (i) of high index at \((\alpha,\ldots,\alpha)\), but (ii) of low index at \(P = (h_1/q_1,\ldots,h_m/q_m)\). A certain partial derivative of \(R\), taken at \(P\), would then, by (i), be very small in absolute value, while, by (ii), it would be \(\ne 0\). As \(R\) has a rational value of denominator \(q_1^{r_1}\cdots q_m^{r_m}\), one would arrive at a contradiction.

“If \(\alpha\) is an algebraic irrational number, and if there are infinitely many fractions \(h/q\) with \(\vert \alpha - h/q\vert \le q^{-\kappa}\), \(q > 0\), \((h, q) = 1\), then \(\kappa\le 2\).”

Much of the proof runs on classical lines. The new ideas are concerned with the multiple zeros at rational points of polynomials in many variables. If \(R(x_1,\ldots,x_m)\not\equiv 0\) is such a polynomial, and if \(\alpha_1,\ldots, \alpha_m\) are real numbers and \(r_1,\ldots, r_m\) are positive integers, write

\[ R(\alpha_1+ y_1, \ldots, \alpha_m+ y_m) = \sum_{j_1=0}^\infty \cdots \sum_{j_m=0}^\infty c(j_1,\ldots, j_m) y_1^{j_1} \cdots y_m^{j_m}. \]

Then the index of \(R\) at \((\alpha_1,\ldots, \alpha_m)\) relative to \(r_1,\ldots, r_m\) is defined as \[ \min (j_1/r_1 + \ldots + j_m/r_m) \] extended over all \(j_1,\ldots, j_m\) with \(c(j_1,\ldots, j_m) \ne 0\). This index is a non-Archimedean (logarithmic) valuation. Next let \(B\), \(r_1,\ldots, r_m\), \(q_1,\ldots, q_m\) be positive integers, and let \(\mathfrak R_m = \mathfrak R_m(B; r_1,\ldots,r_m)\) be the set of all polynomials

\[ R(x_1,\ldots,x_m) = \sum_{k_1=0}^{r_1}\cdots \sum_{k_m=0}^{r_m} a_{k_1\cdots k_m} x_1^{k_1} \cdots x_1^{k_1} \not\equiv 0 \]

with integral coefficients satisfying \(\vert a_{k_1\cdots k_m}\vert \le B\). Denote by \(\Theta_m(B; q_1,\ldots,q_m; r_1,\ldots,r_m)\) the upper bound of the index of \(R\) at the point \((h_1/q_1, \ldots, h_m/q_m)\), extended over all \(R\in\mathfrak R_m\) and over all integers \(h_1,\ldots, h_m\) prime to \(q_1,\ldots, q_m\), respectively. The main lemma states:

“Let \(m\), \(r_1,\ldots, r_m\) , \(q_1,\ldots, q_m\) be positive integers and \(\delta\) a real number such that \(0 < \delta <1/m\), \(r_m > 10/\delta\), \(r_{j-1}/r_j> 1/\delta\) for \(j = 2, 3,\ldots, m\), \(\log q_1 > m (2m + 1)/\delta\), \(r_j\log q_j \ge r_1 \log q_1\) for \(j = 2, 3,\ldots, m\). Then

\[ \Theta_m\left(q_1^{\delta r_1}, q_1,\ldots, q_m; r_1,\ldots, r_m\right) < 10^m \delta^{(1/2)^m}.'' \]

This lemma is proved by induction for \(m\).

If the theorem were false, one could select \(m\) solutions \(h_j/q_j\) of \(\vert \alpha - h_j/q_j\vert < q_j^{-\kappa}\), where \(\kappa > 2\), and construct a polynomial \(R\ne 0\) with not too large integral coefficients, which would be (i) of high index at \((\alpha,\ldots,\alpha)\), but (ii) of low index at \(P = (h_1/q_1,\ldots,h_m/q_m)\). A certain partial derivative of \(R\), taken at \(P\), would then, by (i), be very small in absolute value, while, by (ii), it would be \(\ne 0\). As \(R\) has a rational value of denominator \(q_1^{r_1}\cdots q_m^{r_m}\), one would arrive at a contradiction.

Reviewer: Kurt Mahler (Manchester)

### MSC:

11J68 | Approximation to algebraic numbers |